Calculus PYQ — Page 5

The nearest integral value of the shaded area shown below is: 

The minimum value of the function $f(x) = x^3 + (10-x)^3$ occurs at:

The maximum value of the function $f(x) = x^2(60 - x)$ in [20, 80] is:

The maximum value of $f(x) = \frac{1}{4x^2 + 2x + 1}$ is

The maximum value of f(x) = $\left(\frac{1}{x}\right)^x$ is

The maximum value of $\left(\frac{1}{x}\right)^x$ for $x > 0$ is

The marginal cost of production of x units of a commodity is $56 + \frac{3}{2}x$. It is known that fixed costs are Rs.115. Then the total cost of producing 50 units is:-

The marginal cost (MC) and marginal revenue (MR) functions of a product are $MC = 20 + \frac{x}{20}$ and $MR = 30$ respectively. If the fixed cost is 200, then the maximum value of the profit is:

The length of a rectangle is decreasing at the rate of 4 cm/minute and the width is increasing at the rate of 3 cm/minute, then the rate of change of the perimeter is

The largest open interval, in which the function $f(x) = \frac{x}{x^2 + 1}$ increases, is

The largest open interval in which the function $f(x) = 4x^3 - 5x^2 - 8x + 12$ increases, is:

The largest interval, in which the function $f(x) = x^3 + 2x^2 - 1$ is increasing, is:

The interval(s), where the function $f(x) = \begin{cases} \frac{1-e^x}{e^{2x}-1} & : x \neq 0 \\ \frac{-1}{2} & : x = 0 \end{cases}$ is increasing, is/ are:

The interval on which the function $f(x) = x^4 - \frac{x^3}{3}$ is strictly decreasing, is:

The interval, on which the function $f(x) = x^2e^{-x}$ is increasing, is equal to

The interval on which the function $f(x) = x^3 + 2x^2 - 1$ is decreasing, is

The interval in which the function $f(x) = 2x^3 + 3x^2 - 12x + 1$ is strictly increasing, is

The interval in which the function $g(x) = x^2 e^{-x}$ is increasing is:

The integrating factor of the differential equation, $x^2 \frac{dy}{dx} + xy = log_e x$ is equal to

The integrating factor of the differential equation $(x log_e x)\frac{dy}{dx} + y = 2log_e x$ is

The integrating factor of the differential equation $\frac{dy}{dx} = x + xy$ is

The integral $\int \frac{2dx}{e^{2x}-1}$ is equal to:

The integral $\int e^x\left(\frac{x-1}{2x^2}\right)dx$ is equal to

The integral I = $\int \frac{e^{5\log_e x} - e^{4\log_e x}}{e^{3\log_e x} - e^{2\log_e x}} dx$ is equal to